Given a vector drawn on a two dimensional coordinate axis, how does one calculate the componant of the vector projected along the x-axis?

A vector is a physical quantity which has not only a size,
but also a direction.  For example: to walk 30 meters North-East is a vector quantity
(displacement) because it includes a size (30 m) and a direction (North-East). 
Quantities, such as mass, which do not require a direction are called
scalars.

We can represent a vector by making a scale
drawing on a coordinate plane.  When it is possible to do so, we pick the origin of the
coordinate plane to be where the vector begins.  The vector is then drawn using an arrow
that has a length proportional to the size of the vector and is pointing in the same
direction as the vector. 

For example: if the above vector
is drawn on standard 1/4" graph paper and each quarter of an inch represents 1 meter,
then the length of the arrow on would be 7.5 inches and would be pointing 45 degrees up
from the x-axis.

Any vector can be represented by a pair of
horizontal (x-axis) and vertical (y-axis) components.  The two component represents the
straight line distance in the horizontal and vertical directions one would have to
measure from the beginning of the vector to reach the end of the vector.  When these two
vectors are drawn on the same coordinate axis as the vector they create the two legs of
a right triangle and the vector creates the hypotenuse of the right
triangle.

Using simple right-angle trigonometry we can
calculate the length of each component if we know the original lenght of the vector and
the angle it makes to the x-axis.  From trig we
know

`costheta =
x/(hypotenuse)`   

`sintheta =
y/(hypotenuse)`

Recalling that the hypotenuse is also
the length of the original vector, the x-component and y-components would
be

`x = Vcostheta`  

 `y =
Vsintheta`

 Using the example above the x-axis component
would be

`x = 30m* cos(45) = 21.2
m`

and the y-axis component would
be

`y = 30m*sin(45) = 21.2m`